Theorem sseqn 11228

Description: Two ways to express the subsets of a class of a given size. It might seem that { x e. ~P A | (♯ ` x ) = N } would suffice, but that would require the converse of hashcl 11169 or something similar. Although each side of the equality would be well defined if we changed N e. NN0 to N e. ZZ, they would give different results for the (degenerate) case of a negative size, as shown at ssenneg 11229 and sshashneg 11230. (Contributed by Jim Kingdon, 22-May-2026.)

Assertion

Ref	Expression
sseqn	|- ( N e. NN0 -> { x e. ~P A | x ~~ ( 1 ... N ) } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } )

Distinct variable group:   x, N

Allowed substitution hint:   A( x)

Proof of Theorem sseqn

Step	Hyp	Ref	Expression
1		1zzd 9621	. . . . . . . 8 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> 1 e. ZZ )
2		simpll 527	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> N e. NN0 )
3	2	nn0zd 9716	. . . . . . . 8 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> N e. ZZ )
4	1, 3	fzfigd 10817	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> ( 1 ... N ) e. Fin )
5		enfii 7142	. . . . . . 7 |- ( ( ( 1 ... N ) e. Fin /\ x ~~ ( 1 ... N ) ) -> x e. Fin )
6	4, 5	sylancom 420	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> x e. Fin )
7		simpr 110	. . . . . . . 8 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> x ~~ ( 1 ... N ) )
8		hashen 11172	. . . . . . . . 9 |- ( ( x e. Fin /\ ( 1 ... N ) e. Fin ) -> ( (♯ ` x ) = (♯ ` ( 1 ... N ) ) <-> x ~~ ( 1 ... N ) ) )
9	6, 4, 8	syl2anc 411	. . . . . . . 8 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> ( (♯ ` x ) = (♯ ` ( 1 ... N ) ) <-> x ~~ ( 1 ... N ) ) )
10	7, 9	mpbird 167	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> (♯ ` x ) = (♯ ` ( 1 ... N ) ) )
11		hashfz1 11171	. . . . . . . 8 |- ( N e. NN0 -> (♯ ` ( 1 ... N ) ) = N )
12	2, 11	syl 14	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> (♯ ` ( 1 ... N ) ) = N )
13	10, 12	eqtrd 2267	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> (♯ ` x ) = N )
14	6, 13	jca 306	. . . . 5 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ x ~~ ( 1 ... N ) ) -> ( x e. Fin /\ (♯ ` x ) = N ) )
15		simprr 533	. . . . . . . 8 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ ( x e. Fin /\ (♯ ` x ) = N ) ) -> (♯ ` x ) = N )
16	15	oveq2d 6074	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ ( x e. Fin /\ (♯ ` x ) = N ) ) -> ( 1 ... (♯ ` x ) ) = ( 1 ... N ) )
17		isfinite4im 11180	. . . . . . . 8 |- ( x e. Fin -> ( 1 ... (♯ ` x ) ) ~~ x )
18	17	ad2antrl 490	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ ( x e. Fin /\ (♯ ` x ) = N ) ) -> ( 1 ... (♯ ` x ) ) ~~ x )
19	16, 18	eqbrtrrd 4138	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ ( x e. Fin /\ (♯ ` x ) = N ) ) -> ( 1 ... N ) ~~ x )
20	19	ensymd 7036	. . . . 5 |- ( ( ( N e. NN0 /\ x e. ~P A ) /\ ( x e. Fin /\ (♯ ` x ) = N ) ) -> x ~~ ( 1 ... N ) )
21	14, 20	impbida 600	. . . 4 |- ( ( N e. NN0 /\ x e. ~P A ) -> ( x ~~ ( 1 ... N ) <-> ( x e. Fin /\ (♯ ` x ) = N ) ) )
22	21	pm5.32da 452	. . 3 |- ( N e. NN0 -> ( ( x e. ~P A /\ x ~~ ( 1 ... N ) ) <-> ( x e. ~P A /\ ( x e. Fin /\ (♯ ` x ) = N ) ) ) )
23		elin 3406	. . . . 5 |- ( x e. ( ~P A i^i Fin ) <-> ( x e. ~P A /\ x e. Fin ) )
24	23	anbi1i 458	. . . 4 |- ( ( x e. ( ~P A i^i Fin ) /\ (♯ ` x ) = N ) <-> ( ( x e. ~P A /\ x e. Fin ) /\ (♯ ` x ) = N ) )
25		anass 401	. . . 4 |- ( ( ( x e. ~P A /\ x e. Fin ) /\ (♯ ` x ) = N ) <-> ( x e. ~P A /\ ( x e. Fin /\ (♯ ` x ) = N ) ) )
26	24, 25	bitri 184	. . 3 |- ( ( x e. ( ~P A i^i Fin ) /\ (♯ ` x ) = N ) <-> ( x e. ~P A /\ ( x e. Fin /\ (♯ ` x ) = N ) ) )
27	22, 26	bitr4di 198	. 2 |- ( N e. NN0 -> ( ( x e. ~P A /\ x ~~ ( 1 ... N ) ) <-> ( x e. ( ~P A i^i Fin ) /\ (♯ ` x ) = N ) ) )
28	27	rabbidva2 2799	1 |- ( N e. NN0 -> { x e. ~P A | x ~~ ( 1 ... N ) } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {crab 2526   i^i cin 3213   ~Pcpw 3674   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   ~~ cen 6986   Fincfn 6988   1c1 8144   NN0cn0 9513   ...cfz 10361  ♯chash 11163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-ihash 11164

This theorem is referenced by: (None)